Change in internal energy and enthalpy of Non-spinning black holes in XRBs

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1 Change in internal energy and enthalpy of Non-spinning black holes in XRBs Tanveer Ali Khan, Dipo Mahto, Raj Kumar Sah University Department of Physics, T.M.B.U. Bhagalpur , India Department of Physics, Marwari College, T.M.B.U. Bhagalpur , India Department of Physics, S.S.V. College, Kahalgoan, India Abstract In the present paper, we have derived an expression for the change in internal energy and enthalpy of Non -spinning E K R black holes using the relation( BH BH s S K R ) as proposed by Kanak Kumari et al.,2010 and entropy bh 2 BH bh change( c ) as proposed by Mahto et al., 2014 and showed that the change in mass, internal energy and enthalpy of Non-spinning black holes are the same. Thereafter, we have calculated their values for d ifferent test Non-spinning black holes existing in XRBs. Keywords Surface gravity, Enthalpy and Non-spinning black holes and XRBs. Academic Discipline and Sub-Disciplines Astrophysics, thermodynamics and relativity. SUBJECT CLASSIFICATION Physics Classification; TYPE (METHOD/APPROACH) This work is completely theoretical based on the analysis of laws of thermodynamics and black hole mecha nics and concluded that the change in mass, internal energy and enthalpy of Non-spinning black holes are manifestation of the same thing in XRBs. INTRODUCTION A black hole is a vacancy with strong gravitation and one of the most remarkable achievements in gravitational physics having temperature and entropy[1, 2]. The surface gravity has the same role in the black hole mechanics as the temperature in the ordinary laws of thermodynamics [2], while the entropy is not only an intrinsic property of a given system and is a function of the energy and volume, but gives instead a count of the total number of microscopic states available to a macroscopic system that has fixed the total energy and charge[2]. The enthalpy is heat content of homogenous system[3] which is related to some thermodynamic parameters like energy and entropy of the same system. Classically, the black holes are perfect absorbers and do not emit anything, their temperature is absolute zero, however, in quantum theory black holes emit Hawking radiation with a perfect thermal spectrum[3,4]. In 1912, Dolan, B.P. discussed the pressure and enthalpy of black holes using first law of thermodynamics with positive and negative cosmological constant[5]. In 1913, Mahto et al. derived an expression for the change in internal energy and enthalpy of black holes using the basic concept of entropy and first law of thermodynamics [3]. In the present work, we have proposed an expression for the change in internal energy and enthalpy of Non -spinning E K R black holes using the relation ( BH BH s S 2 K R / c 2 ) as proposed by Kanak Kumari et al.,2010 and entropy change( ) as proposed by Mahto et al., 2014 and shown that the change in mass, internal energy and enthalpy of Non-spinning black holes are manifestation of the same thing. We also have calculated their values for different test Non-spinning black holes existing in X-ray binaries (XRBs). DISCUSSION There are some systems which may show no apparent mechanical energy, but may still be capable of doing work. They are said to possess internal energy [6]. 234 P a g e e d i t o c p u b l i s h i n g. c o m

2 From first law of thermodynamics, we have Q du PdV...(1) From second law of thermodynamics, we have Q TS......(2) From equation(1) and (2), we obtain Q du PdV...(3) The enthalpy of a homogenous system is defined as H U PV...(4) Differentiating equation(4), we have dh du PdV VdP...(5) Subtracting the equation(5) from equation(3) and solving, we have TS dh VdP Or dh T S VdP...(6) The equation(6) shows the change in enthalpy of the system takes place with corresponding change in the entropy and pressure of the system that we have taken into consideration. In the case black hole mechanics, the term / plays the role of temperature of black holes, hence the equation(6) may be written as: dh S VdP...(7) If the pressure of black holes is supposed to have constant pressure, then dp=0, and hence the equation(7) can be written as: dh S...(8) The first law of black hole mechanics is simply an identity relating the change in mass M, angular momentum J, horizon area A and charge Q of a black hole. The first order variations of these quantities in the vacuum satisfy[7,8,9]. M A J Q 8...(9) 235 P a g e e d i t o c p u b l i s h i n g. c o m

3 Where =Angular velocity of the horizon, v=difference in the electrostatic potential between infinity and horizon. Since the event horizon A of black hole is related entropy S as : S A 4 Or 4...(10) A S...(11) Or A 4S...(12) Putting the above value in equation(9), we have M S J Q...(13) For Non-spinning black holes, J=0 and hence dj=0, which leads the above equation as given below:[2,12] M S Q...(14) Actually an astronomical black hole is not likely to have any significant electric charge, hence dq=0,[12] which leads the equation(14) M S...(15) When equation(8) is compared with the equation(15), we obtain dh M...(16) We have earlier shown that the change in internal energy and enthalpy are the manifestation of the same thing at constant pressure and volume[3]. i.e. du dh...(17) From equation(16) and (17), we have dm du dh...(18) The equation(18) shows that the change in mass, internal energy and enthalpy of Non -spinning black holes are the manifestation of the same thing at constant volume and pressure if required. From equation(8), we have dh S...(19) 236 P a g e e d i t o c p u b l i s h i n g. c o m

4 The change in entropy of non-spinning black holes for the corresponding change in energy is given by the following equation[7] S bh E 2 c...(20) The energy of non-spinning black holes in terms of the radius of event horizon is given as [10]. E K R K R BH BH s BH bh...(21) Where KBH / J m...(22) The equation(21) is differentiated, we have E K R BH BH bh...(23) Putting equation(23) in equation(20), we have S K R bh 2 BH bh c...(24) Substituting the above value in equation(6.19), we get H K R...(25) U K R Or...(26) (Taking the velocity of light is unity, i.e c=1). From equation (23) and (25), we have E BH H BH...(27) The above equation shows that the change in energy and enthalpy of non-spinning black holes are the same. In fact, this energy is internal energy. DATA IN SUPPORT FOR MASS OF SUN, BLACK HOLES AND BLACK HOLE CONSTANT: There are two categories of black holes classified on the basis of their masses clearly very distinct from each other, with very different masses M ~ 5 20 Mʘ for stellar mass Black holes in X-ray binaries and M ~ Mʘ for super massive black holes in Active Galactic Nuclei[2,12]. Mass of sun (Mʘ) = 1.99 x Kg, Black hole const. (KBH)= 43 J/K P a g e e d i t o c p u b l i s h i n g. c o m

5 TABLE: 1 GRAPH 1: Change in Internal energy and Enthalpy of Non-spinning black holes in XRBs. Sl.No Mass of BHs (M) (Rs=2950 (in metre) M/Mʘ) Change in internal energy of black holes (dubh) in Joule. Change in enthalpy of black holes ( H 1 5Mʘ x x M ʘ x x Mʘ x x Mʘ x x Mʘ x x Mʘ x x Mʘ x x Mʘ x x M ʘ x x M ʘ x x Mʘ x x M ʘ x x Mʘ x x Mʘ x x Mʘ x x Mʘ x x10 48 bh ) in Joule (Fig 1: The graph plotted betn the radius of event horizon and corresponding change in enthalpy/internal energy of nonspinning black hole in XRBs) 238 P a g e e d i t o c p u b l i s h i n g. c o m

6 DISCUSSION AND RESULT: In present work, we have derived the formula for the change in internal energy and enthalpy of Non-spinning black holes U K R H K R represented by eqn. Thus we see that the change in internal energy and enthalpy of Non-spinning black holes are the same. We also have observed that the change in internal energy and enthalpy of Non-spinning black holes are exactly the same to the change in mass and energy of Non -spinning black holes as clear from equation(23), (25) and (26). These relations show that the change in internal energy, enthalpy, mass and energy Non-spinning black holes are the manifestation of the same thing. After showing these facts, we also have calculated their values for different test Non-spinning black holes in X-ray binaries (XRBs) and then graphs have been plotted between the radius of event horizon (R bh) of different test black holes and their corresponding values of change in enthalpy in XRBs (fig. 1 ). and The graph plotted between the radius of event horizon (R bh) of different test black holes and their corresponding values of change in internal energy as well as change in energy will be also the same in XRBs as clear from the expression (25), (26) and (27). From the observation of the data in the table1 and graphs plotted in fig1, it is clear that the change in internal energy as well as change in enthalpy of the Non-spinning black holes increases linearly with increase of mass/radius of the event horizon of the black holes in both X-ray binaries (XRBs). It is also clear that the change in internal energy as well as change in enthalpy of the Non-spinning black holes are increasing with a constant rate with the mass of the black holes/the radius of event horizon in XRBs. The graph plotted between the radius of the event horizon and corresponding change in internal energy and enthalpy of black holes are in straight line in both the X-ray binaries (XRBs). U K R H K R The straight line nature of the graph shows the validity of the relations. The straight line also shows that there is a definite relation between the mass/the radius of the event horizon and change in internal energy as well as enthalpy of the black holes. CONCLUSION: In the present research work, we have drawn the following conclusions: and 1. The change in internal energy, enthalpy, mass and energy of the Non-spinning black holes are the manifestation of the same thing. 2. The expression for the change in internal energy and enthalpy of black hole can be used to determine the change in internal energy and enthalpy of different types of test Non-spinning black holes in XRBs. 3. The black hole constant for non-spinning black holes is x Jm -1 which has vital role in calculation of change in internal energy and enthalpy of Non-spinning black holes in XRBs. 4. There is a uniform variation between the change in internal energy and enthalpy of non -spinning black holes with the radius of the event horizon of different types of the test non-spinning black holes in XRBs. U K R 5. The straight line nature of graph for XRBs gives the validity of relations H K R. 6. Larger the mass/ radius of event horizon of the non-spinning black holes, greater is the change in internal energy as well change in enthalpy of non-spinning black holes. REFERENCES: 1. Zhao, H.; Li.H; Hu.S; Zhao, R Geometrical uncertainty principle and black hole entropy of higher dimensional de-sitter space time. Commun. Theor. Phys Mahto, D.,Prakash.V.,Prasad.U; Singh, B.K., Singh, K.M Change in entropy of non-spinning black holes w.r.t. the radius of event horizon in XRBs. Astrophysics and Space Science. 3. Mahto, D., Singh, A.K., Ram, M. Vineeta, K Change in internal energy and enthalpy of black holes. International Journal of Astrophysics and Space Science, USA, 4. Wald, R.M The thermodynamics of black holes. Living reviews in relativity. 5. Dolan, B. P Pressure and volume in the first law of black hole thermodynamics. Class.Quantum Grav., 28:235017, [arxiv: ]. 6. Saha. M. N., & Shrivastava,B. N A treatise on Heat. The Indian Press Private Ltd. Allahabad. 7. Mahto, D., Kumari, K., Sah, R.K., Singh, K.M Study of Non-spinning black holes with reference to the change in Energy and Entropy. Astrophys Space Sci Doi /s and 239 P a g e e d i t o c p u b l i s h i n g. c o m

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